By Mikhail Gromov, Jacques LaFontaine, Pierre Pansu, S. M. Bates, M. Katz, P. Pansu, S. Semmes

ISBN-10: 0817645829

ISBN-13: 9780817645823

ISBN-10: 0817645837

ISBN-13: 9780817645830

According to "Structures Mtriques des Varites Riemanninnes", edited through J. LaFontaine & P. Pansu.

Metric conception has gone through a dramatic part transition within the final many years whilst its concentration moved from the rules of genuine research to Riemannian geometry and algebraic topology, to the idea of countless teams and likelihood conception. the hot wave all started with seminal papers by way of Svarc and Milnor on progress of teams and the magnificent facts of the pressure of lattices via Mostow. This was once through the production of the asymptotic metric conception of countless teams by way of Gromov.

The structural metric method of the Riemannian classification, tracing again to Cheegers thesis, pivots round the suggestion of the Gromov-Hausdorff distance among Riemannian manifolds. This distance organizes Riemannian manifolds of all attainable topological forms right into a unmarried hooked up moduli house, the place convergence permits the cave in of size with unexpectedly wealthy geometry, as printed within the paintings of Cheeger, Fukaya, Gromov and Perelman. additionally, Gromov came across metric constitution inside of homotopy thought and therefore brought new invariants controlling combinatorial complexity of maps and areas, reminiscent of the simplicial quantity, that is answerable for levels of maps among manifolds. through the related interval, Banach areas and likelihood conception underwent a geometrical metamorphosis, encouraged through the Levy-Milman focus phenomenon, encompassing the legislation of huge numbers for metric areas with measures and dimensions going to infinity.

The first levels of the hot advancements have been offered in a direction introduced by way of Gromov in Paris, and become the recognized 1979 eco-friendly e-book by means of LaFontaine and Pansu. the current English translation of that paintings has been enriched and improved with new fabric to mirror fresh growth. also, 4 appendices---by Gromov on Levys inequality, by way of Pansu on quasiconvex domain names, by way of Katz on systoles of Riemannian manifolds, and through Semmes overviewing research on metric areas with measures---as good as an in depth bibliography and index around out this special and gorgeous booklet.

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**Extra resources for Metric Structures for Riemannian and Non-Riemannian Spaces**

**Example text**

12^ Remark: The latter argument apphes to an arbitrary (say, compact for safety) metric space X admitting a topological fibration p : X -^ B, all of whose fibers are circles of length < 27r. , / i = fo ^ P- Now let us suppose that the fibers p~^{b) C X are /c-dimensional and that their /c-dimensional Hausdorfl^ measures are strictly smaller than that of the unit sphere S^. Here, one wants to homotope each map f : X -^ S^ with dil(/) < 1 to / i , sending each fiber p~^{b) onto a subset of topological (or Hausdorff") dimension < fc in 5^.

20. The requirements for a map to be an isometry or even local isometry are too stringent to provide a sufficiently rich class of morphisms for path metric spaces. For example, any Riemannian n-manifold that is locally isometric to R^ must be flat. A more flexible notion is the following. 21. Definition: An arc-wise isometry of path metric spaces X,Y is a map f:X-^Y such that i{f o c) = £{c) for each Lipschitz curve c: I -^ X oiX. Examples: (1) Every closed, piecewise C^ curve admits an arc-wise isometric mapping into R.

The proportionality constant equals the degree of f defined above. (See [Massey] and [Godb], p. 212). 7. Remark: If M is a compact, oriented Riemannian manifold, then its volume form is defined as the unique differential n-form VM , which for every x e M has value 1 on each oriented, orthonormal frame in T^M. The volume of M equals the integral vol(M) = J^^ UM- If N is another compact, oriented Riemannian manifold with volume form z/^v, and / : M -^ iV is smooth, then deg(/) = {YO\{N))~^ J^j f^u^.

### Metric Structures for Riemannian and Non-Riemannian Spaces by Mikhail Gromov, Jacques LaFontaine, Pierre Pansu, S. M. Bates, M. Katz, P. Pansu, S. Semmes

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